1. **State the problem:** We need to resolve the given rational function into partial fractions. 2. **General formula and rules:** For a rational function $\frac{P(x)}{Q(x)}$ where the degree of $P(x)$ is less than the degree of $Q(x)$, and $Q(x)$ can be factored into linear or irreducible quadratic factors, we express it as a sum of simpler fractions. 3. **Example:** Suppose the function is $\frac{1}{(x-1)(x+2)}$. We write: $$\frac{1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$$ 4. **Multiply both sides by the denominator:** $$1 = A(x+2) + B(x-1)$$ 5. **Expand and collect terms:** $$1 = Ax + 2A + Bx - B = (A+B)x + (2A - B)$$ 6. **Equate coefficients:** For $x$: $0 = A + B$ For constant: $1 = 2A - B$ 7. **Solve the system:** From $0 = A + B$, we get $B = -A$ Substitute into $1 = 2A - B$: $$1 = 2A - (-A) = 3A \Rightarrow A = \frac{1}{3}$$ Then $B = -\frac{1}{3}$ 8. **Final answer:** $$\frac{1}{(x-1)(x+2)} = \frac{1/3}{x-1} - \frac{1/3}{x+2}$$ This method applies similarly to other rational functions once factored.